Method for localising at least one emitter

ABSTRACT

The invention relates to a method of localization of one or more sources, said source or sources being in motion relative to a network of sensors, the method comprising a step of separation of the sources in order to identify the direction vectors associated with the response of the sensors to a source having a given, incidence, characterized in that it comprises at least the following steps: 
         associating the direction vectors a 1m  . . . a Km  obtained for the m th  transmitter and respectively at the instants t 1  . . . t K ,    localizing the mth transmitter from the associated vectors a 1m  . . . a Km .

The invention relates to a method for the localization of one or more transmitters.

It can be applied essentially to the localization of transmitters on the ground from a moving vehicle without a priori knowledge concerning the signals sent.

The technical field in particular is that of the passive localization of transmitters.

FIG. 1 illustrates an airborne localization. The transmitter is at the position (x₀,y₀,z₀). The carrier at the instant t_(k) is at the position (x_(k),y_(k),z_(k)) and perceives the transmitter at the angles of incidence (θ(t_(k),x₀,y₀,z₀), Δ(t_(k),x₀,y₀,z₀)). The angles θ(t,x₀,y₀,z₀) and Δ(t,x₀,y₀,z₀) evolve in time and depend on the position of the transmitter as well as the trajectory of the carrier.

The angles θ(t,x₀,y₀,z₀) and Δ(t,x₀,y₀,z₀) are determined relative to a network of N antennas that can be fixed beneath the carrier as shown in FIG. 2.

There are several existing techniques to determine the position (x_(m), y_(m), z_(m)) of a transmitter. These techniques of localization differ especially in the parameters which are estimated instantaneously at the level of the network of sensors. They can be classified as follows.

Use of Goniometry

These techniques are known and used in the prior art. In most cases, they are based on a 1D azimuthal goniometry. The azimuths θ_(km)=θ(t_(k),x_(m),y_(m),z_(m)) associated with the m^(th) transmitter are measured for different instants t_(k). In using the position (x_(k),y_(k),z_(k)) of the carrier at the corresponding instant k, a position (x_(mk),y_(mk),z_(mk)) of the transmitter m is estimated by a ground intersection. The position (x_(k),y_(k),z_(k)) of the carrier is given by a GPS unit, its orientation is obtained by a compass in the case of a ground carrier and by an inertial navigation system in the case of an aircraft. From all the positions (x_(mk),y_(mk),z_(mk)), the method extracts data with which it is possible to determine the M dominant positions (x_(m),y_(m),z_(m)) of the incident transmitters. The localization is obtained by triangulation or by ground intersection (2D goniometry). The drawback of triangulation techniques is that they require major movement. Furthermore, goniometry techniques require the use of a network of non-ambiguous sensors. This has the drawback of necessitating a calibration table and restricting the size of the sensor network and therefore giving incidence values of limited precision.

Use of the Phase Difference Between Two Remote Sensors

The inter-sensor phase difference Δφ(t_(k),x₀,y₀,z₀) depends on the positions of the two sensors as well as the incidence θ(t_(k),x₀,y₀,z₀), Δ(t_(k),x₀,y₀,z₀)) of the transmitter. This phase, which depends on time, is directly related to the position (x₀,y₀,z₀) of the transmitter. Consequently, studying the function of time Δφ(t,x₀,y₀,z₀) makes it possible to deduce the position (x₀,y₀,z₀) of the transmitter. In this group of applications, the two sensors are distant in order to augment the precision of measurement of the phase. This has the drawback of causing variations in the phase difference Δφ(t,x₀,y₀,z₀) as a function of time over more than 2π and the technique then necessitates a step enabling the phase to be made to vary by more than 2π. Furthermore, in this technique, the phase is measured by carrying out a direct intercorrelation between two sensors, and cannot be used to deal with the multiple-sensor case.

Use of the Measurement of the Carrier Frequency of the Transmitter

These techniques make use of the fact that the estimated carrier frequency is the sum of the carrier frequency of the transmitter and the Doppler shift due to the speed of movement of the carrier. The Doppler shift has the advantage of depending on the position (x₀,y₀,z₀) of the transmitter and of also being a function of the time Δf(t,x₀,y₀,z₀). Consequently, studying the function of the time Δf(t,x₀,y₀,z₀) makes it possible to deduce the position (x₀,y₀,z₀) of the transmitter therefrom. However, the measurement of this Doppler shift has the drawback of necessitating transmitters with particular waveforms. This measurement of frequency can be done by cyclical techniques in which it is assumed that the signal sent is non-circular.

Use of Propagation Times

These techniques use the differences in propagation time between antennas (TDOA or time difference of arrival) which are directly related to the respective differences between the transmitter and the different antennas and therefore to the position (x₀,y₀,z₀) of the transmitter. The use of at least three antennas that are sufficiently spaced out enables the position (x₀,y₀,z₀) of the transmitter to be deduced by hyperbolic localization. The drawback of these techniques is that they cannot be implemented in a single-carrier context owing to the considerable spacing required between antennas. Furthermore, in these techniques, the time difference is measured by the direct performance of an inter-correlation between two sensors. This approach cannot be used to deal with the case involving multiple transmitters.

The method of the invention relies especially on a novel approach for the direct estimation of the positions (x_(m), y_(m), z_(m)) of each of the transmitters from a parametric analysis of the multipath signal at various instants t_(k) on a duration Dt. The parametric analysis especially has the additional function of separating the different transmitters at each point in time t_(k). Then, the parameters of a same transmitter coming from the different points in time t_(k) are associated to finally localize each of the transmitters.

The invention relates to a method of localization of one or more sources, said source or sources being in motion relative to a network of sensors, the method comprising a step of separation of the sources in order to identify the direction vectors associated with the response of the sensors to a source having a given angle of incidence. The method is characterized in that it comprises at least the following steps:

-   -   associating the direction vectors a_(1m) . . . . a_(Km) of the         m^(th) transmitter obtained respectively at the instants t₁ . .         . t_(K),     -   localizing the m^(th) transmitter from the associated vectors         a_(1m) . . . . a_(Km).

The method of the invention has especially the following advantages:

-   it enables the localization of the position of the transmitter in     (x, y, z) coordinates and, in addition, that of its speed vector, -   it can be applied when there are one or more incident transmitters     present, -   its implementation does not require any particular knowledge of the     signal sent, -   it enables the use of a network of ambiguous sensors, i.e. several     incidence values are associated with the same response of the     network which have the advantage of being great and therefore of     being more robust with respect to inter-antenna coupling phenomena     or, more generally, antenna-network modeling errors, -   it can be implemented in networks calibrated in (θ,Δ), -   it can be implemented in networks with amplitude-diversity antennas     such as co-localized antennas: networks with dipoles having the same     phase center and different orientations.

Other features and advantages of the object of the present invention shall appear more clearly from the following description given by way of an illustration that in no way restricts the scope of the invention and from the appended figures, of which:

FIG. 1 shows the principle of the localization by an aircraft of a transmitter positioned on the ground,

FIG. 2 shows the relationship between a network of antennas and the value of incidence of a transmitter,

FIG. 3 is a general drawing explaining the working of the method according to the invention,

FIGS. 4, 5 and 6 are examples of implementation of the method according to the invention.

In order to provide for a clearer understanding of the object of the present invention, the following description is given by way of an illustration that in no way restricts the scope of the invention for the localizing of several transmitters positioned on the ground using a network of sensors installed in an aircraft in motion. Such a system is described for example in FIG. 1. The aircraft is equipped with a processor adapted to implementing the steps of the method according to the invention.

The method can also be implemented in the context of vehicles moving on the ground.

FIG. 3 is a representation, in a signal time/amplitude graph, of the signal x(t) formed by a combination of the signals of the transmitters at different instants t₁, t₂, . . . t_(K). This figure summarizes the different steps implemented, namely the separation of the transmitters SE and the parametric estimation EP, the association of the parameters of each transmitter and the localization of a transmitter.

In the presence of M transmitters, at the instant t at output from the N sensors of the network, the method has the vector x(t) representing the mixture of signals from the M transmitters. Around the instant t_(k), the vector x(t+t_(k)) sized N×1, representing the mixture of the signals from the M transmitters, is expressed as follows: $\begin{matrix} \begin{matrix} {{x\left( {t + t_{k}} \right)} = {{\sum\limits_{m = 1}^{M}{{a\left( {\theta_{km},\Delta_{km}} \right)}\quad{s_{m}\left( {t + t_{k}} \right)}}} + {b\left( {t + t_{k}} \right)}}} \\ {= {{{A_{k}\quad{s\left( {t + t_{k}} \right)}} + {{b\left( {t + t_{k}} \right)}\quad{pour}\quad{t}}} < {\Delta\quad{t/2}}}} \end{matrix} & (1) \end{matrix}$ where b(t) is the noise vector assumed to be Gaussian, a(θ,Δ) is the response of the network of sensors to a source having an incidence (θ,Δ), A_(k)=[a(θ_(k1), Δ_(k1)). . . a(θ_(kM), Δ_(kM))], s(t)=[s₁(t) . . . s_(M)(t)]^(T), θ_(km)=θ(t_(k),x_(m),y_(m),z_(m)) and Δ_(km)=Δ(t_(k),x_(m),y_(m),z_(m)). In this model, the mix matrix A_(k) depends on the instant t_(k) of observation. The direction vector with incidence corresponding to the m^(th) transmitter at the instant t_(k) a _(km) =a(θ_(km), Δ_(km))=a(t _(k) ,x _(m) ,y _(m) ,z _(m)) of the m ^(th) transmitter  (2) is a known function of t_(k) and of the position of the transmitter (x_(m),y_(m),z_(m)).

The method according to the invention comprises at least the following steps:

-   1. estimating one or more parameters associated with the position of     the source, for example the direction vectors, the incidence values,     the position etc. and separating the M transmitters for the     different instants t_(k): this consists in identifying the direction     vectors having an incidence a_(km) for (1≦m≦M). This first step is     performed, for example, by means of source separation techniques     known to those skilled in the art. -   2. associating the parameters estimated for the m^(th) transmitter,     for example in associating the different direction vectors having     incidence values, a_(1m) . . . a_(km) obtained respectively at the     instants t₁, . . . t_(k), -   3. localizing the m^(th) transmitter from the associated vectors.     Association Step

In the presence of M transmitters and after separation of sources, the method, at the instant t_(k), possesses the M signatures a_(km) for (1≦m≦M). At the instant t_(k+1) the source separation gives the M vectors b_(i) for (1≦i≦M). The goal of this tracking is to determine, for the m^(th) transmitter, the index i(m) which minimizes the difference between a_(km) and b_(i(m)). In this case, it will be deduced therefrom that a_(k+i, m)=b_(i(m)). To make this association, the distance between two vectors u and v is defined, for example, by: $\begin{matrix} {{d\left( {u,v} \right)} = {1 - \frac{{{u^{H}\quad v}}^{2}}{\left( {u^{H}\quad u} \right)\quad\left( {v^{H}\quad v} \right)}}} & (3) \end{matrix}$ Where u^(H) is the conjugate transpose of the vector u.

Thus, the index i(m) verifies: $\begin{matrix} {{{d\left( {a_{km},b_{i{(m)}}} \right)} = {\min\limits_{1 \leq i \leq M}\left\lbrack {d\left( {a_{km},b_{i}} \right)} \right\rbrack}}{hence}{{d\left( {a_{km},b_{i{(m)}}} \right)} = {\min\limits_{1 \leq i \leq M}\left\lbrack {1 - \frac{{{a_{k\quad m}^{H}\quad b_{i{(m)}}}}^{2}}{\left( {a_{k\quad m}^{H}\quad a_{k\quad m}} \right)\left( {b_{i{(m)}}^{H}b_{i{(m)}}} \right)}} \right\rbrack}}} & (4) \end{matrix}$

In this association, we consider a function β_(m) associated with the m^(th) transmitter: β_(m)(t_(k))=d(a _(km) , a _(0m))  (5) In the course of the association, for each transmitter m is obtained and for 1≦m≦M, the function β_(m)(t) is obtained. This function is aimed especially at eliminating the instants t_(k) whose value β_(m)(t_(k)) appears to be far too distant from an interpolation of the function β_(m)(t), i.e. the aberrant instants which may be associated with other transmitters are eliminated. A zone of tolerance +/−Δ is defined around the curve defined by the function β_(m)(t_(k)) This zone of tolerance will depend on the precision of estimation of the direction vectors a_(km). In particular, in the presence of M=1 sources, the zone will be in the range of Δ=3/√{square root over (BΔt)}(where Δt is the elementary time of parametric estimation illustrated in FIG. 3 and B is the instantaneous band of the signal x(t)).

The steps of this association for K instants t_(k) are for example the following:

-   Step ASE-1: Initialization of the process at k=2. The initial number     M of transmitters is for example determined by a test to detect the     number of sources at the instant to known t₀ those skilled in the     art, -   Step ASE-2: For 1≦m≦M determining the indices i(m) in applying the     equation (4) and in using the vector a_(k,m) with 1≦m≦M and the     vectors b_(i) identified at the instant t_(k+1) for (1≦i≦M), -   Step ASE-3: For 1≦m≦M performing the operation a_(k+1 m)=b_(i(m)), -   Step ASE-4: Incrementing k←k+1 and if k<K returning to the step     ASE-1, -   Step ASE-5: Starting from the family of instants Φ={t₁<. . . <t_(K)     }, eliminating the/instants t_(i)εΦ such that the coefficients     β_(m)(t_(i)) do not belong to the zone demarcated by the     interpolation curve of the β_(m)(t_(k)) values and the zone of     tolerance Δ. The instants t_(k) where     |β_(m)(t_(k))−β_(m)(t_(k-1))|<Δ will also be eliminated. After this     sorting, the new family of instants is Φ={t₁<. . . <t₁} and we     assume K=1.

At the end of these steps, the method has determined the vectors a_(1m) . . . a_(Km) associated with the m^(th) transmitter.

Localization of a Transmitter

The method determines the position of the m^(th) transmitter from the components of the vectors a_(1m) up to a_(Km). These vectors a_(km) have the particular feature of depending on the instant t_(k) and above all on the position (x_(m),y_(m),z_(m)) of the transmitter. In particular, for a network formed by N=2 sensors spaced out by a distance d in the axis of the carrier, the vector verifies a_(km): $\begin{matrix} {a_{k\quad m} = {\begin{bmatrix} 1 \\ {\exp\left( {j\quad 2\quad\pi\quad\frac{d}{\lambda}\quad{\cos\left( {\theta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)}\quad{\cos\left( {\Delta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)}} \right)} \end{bmatrix} = {a\left( {t_{k},x_{m},y_{m},z_{m}} \right)}}} & (6) \end{matrix}$

The value 1 of the first component corresponds to the reference sensor. According to FIG. 1, the incidence (θ(t_(k),x_(m),y_(m),z_(m)), Δ(t_(k),x_(m),y_(m),z_(m)) ) can be directly computed from the position (x_(k),y_(k),z_(k)) of the carrier at the instant t_(k) and the position (x_(m),y_(m),z_(m)) of the transmitter.

Step of Transformation of the Vector

According to a first alternative embodiment, the method comprises a step of correction of the a_(km), values, the measurement of the direction vectors a_(km) is generally obtained to within an indeterminate complex factor. According to this first variant, the method comprises a step of reducing the phase reference of the direction vector measured in reducing the operation to the phase barycenter (defined to within a constant scalar coefficient that may be set at 1). This operation is performed, for example, by estimating the correction coefficient determined by the following conversion of the values a_(km) into values a′_(km): $\begin{matrix} {a_{k\quad m}^{\prime} = {\left( {\prod\limits_{i}^{\quad}\quad\frac{a_{i}}{a_{i}}} \right)^{- \frac{1}{N}}\quad a_{k\quad m}}} & (7) \end{matrix}$

The correction coefficient is not totally determined by this expression given the Nth order indeterminacy of the complex root. A tracking of the phase evolution during the observation period is therefore done.

Since the complex coefficient is defined to within a factor from among the N N^(th) roots of the unit, the phase tracking consists in arbitrarily setting the first correction coefficient (in taking the root 1 for example), and then in determining, at each new iteration k+1, the coefficient that minimizes the mean phase differences between the direction vector recentered at k+1 and the vector recentered at the instant k.

The minimization criterion for measurement at the same frequency may be equal to: $\begin{matrix} \begin{matrix} {\min\limits_{{\rho ɛ}\sqrt[N]{1}}{\sum\limits_{{\rho ɛ}\quad{value}}^{\quad}{\min\left( {{{mod}\left( {{{\arg\left( \frac{\rho.{a_{k + 1}(i)}}{a_{k}(i)} \right)}},{2\quad\pi}} \right)},} \right.}}} \\ \left. {{2\quad\pi} - {{mod}\left( {\left. {\arg\left( \frac{\rho.{a_{k + 1}(i)}}{a_{k}(i)} \right.} \right),{2\quad\pi}} \right)}} \right) \end{matrix} & (8) \end{matrix}$ where the values a_(k+1) are the direction vectors recentered with the correction coefficient arbitrarily determined by any one of the Nth roots of the expression. For measurement at different frequencies, it is possible to compare the phases of the components of the two direction vectors in correcting them by a power given by the ratio of these two frequencies.

If we consider the vectors b_(km)=a′_(km), it is then possible to compare this measurement with the theoretical value b(t_(k),x_(m),y_(m),z_(m)) for which the theoretical direction vector a(t_(k),x_(m),y_(m),z_(m)) is computed for a theoretical origin at the theoretical (geometrical) phase barycenter (geometrical locus for which the theoretical sum of the phases differences gets cancelled out). This locus (generally) does not coincide with the phase center of the network.

According to another alternative embodiment, the method comprises a step of conversion of the vector a_(km) into a vector b_(km) whose components are formed out of components of the vector a_(km). In particular, the method builds for example the vector b_(km) sized (N−1)×1 in choosing a reference sensor in n=i: $\begin{matrix} {b_{k\quad m} = {\begin{bmatrix} {{a_{k\quad m}(1)}/{a_{k\quad m}(i)}} \\ \vdots \\ {{a_{k\quad m}\left( {i - 1} \right)}/{a_{k\quad m}(i)}} \\ {{a_{k\quad m}\left( {i + 1} \right)}/{a_{k\quad m}(i)}} \\ \vdots \\ {{a_{k\quad m}(N)}/{a_{k\quad m}(i)}} \end{bmatrix} = {b\left( {t_{k},x_{m},y_{m},z_{m}} \right)}}} & (9) \end{matrix}$ where a_(km)(i) is the i^(th) component of a_(km)

The components of b_(km) correspond in this case to the ratios of the components of the vector a_(km) and of the vector a_(km)(i).

Thus in the example of the equation (6) in fixing i=1 we get: $\begin{matrix} {{b_{k\quad m} = {\left\lbrack {\exp\left( {j\quad 2\quad\pi\quad\frac{d}{\lambda}\quad{\cos\left( {\theta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)}\cos\left( {\Delta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)} \right)} \right\rbrack = {{a_{k\quad m}(2)}/{a_{k\quad m}(1)}}}}\quad} & (10) \end{matrix}$

It being known that the direction vectors a_(km) are estimated with a certain error e_(km) such that a_(km)=a(t_(k),x_(m),y_(m),z_(m))+e_(km), it can be deduced therefrom that the same is true for the converted vector b_(km) of (9).

Step of Maximization of a Correlation Criterion

It being known that the vector a_(km) is a function of the position (x_(m),y_(m),z_(m)) of the transmitter, the same is true for the vector b_(km). The method comprises a step of maximization of a normalized vector correlation L_(K)(x,y,z) in the space (x,y,z) of position of a transmitter where $\begin{matrix} {{{L_{K}\left( {x,y,z} \right)} = \frac{{{b_{K}^{H}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}\quad b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}\quad{v_{K}\left( {x,y,z} \right)}} \right)}}{with}{{b_{K} = {\begin{bmatrix} b_{1\quad m} \\ \vdots \\ b_{K\quad m} \end{bmatrix} = {{v_{K}\left( {x_{m},y_{m},z_{m}} \right)} + w_{K}}}},{v_{K} = {\left( {x,y,z} \right) = \begin{bmatrix} {b\left( {t_{1},x,y,z} \right)} \\ \vdots \\ {b\left( {t_{K},x,y,z} \right)} \end{bmatrix}}}}{{{and}\quad w_{K}} = \begin{bmatrix} w_{1m} \\ \vdots \\ w_{K\quad m} \end{bmatrix}}} & (11) \end{matrix}$

The noise vector w_(k) has the matrix of covariance R=E[w_(K) w_(K) ^(H)]. Assuming that the matrix R is known, the criterion can be envisaged with a whitening technique. Under these conditions the following L_(K)′(x,y,z) criterion is obtained: $\begin{matrix} {{{L_{K^{\prime}}\left( {x,y,z} \right)} = \frac{{{b_{K}^{H}\quad R^{- 1}\quad{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}\quad R^{- 1}\quad b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}\quad R^{- 1}\quad{v_{K}\left( {x,y,z} \right)}} \right)}}{{{With}\quad R} = {E\left\lbrack {w_{K}w_{K}^{H}} \right\rbrack}}} & (12) \end{matrix}$

-   It must be noted that the criteria of the equations (11) and (12)     are equal when R=_(σ) ²I, i.e. when the errors are considered to be     of equal level on all the sensors and independent between sensors.     The criterion L_(K)(x,y,z) of the equation (12) is therefore valid     for a white statistics model noise w_(K), -   The criteria L_(K)(x,y,z) and L_(K)′(x,y,z) range between 0 and 1     and verify L_(K)(x,y,z)=L_(K)′(x,y,z)=1 for the position     (x_(m),y_(m),z_(m)) of the m^(th) transmitter. This normalization     enables the setting of an efficient threshold of localization η.     Thus, all the maximum values (x_(m),y_(m),z_(m)) of L_(K)(x,y,z)     which verify L_(K)(x_(m),y_(m),z_(m))>η are considered to be good     localizations. The threshold may be set as a function of an     approximate knowledge of the statistics of w_(K), -   The criteria L_(K)(x,y,z) and L_(K)′(x,y,z) have the advantage of     being capable of implementing a technique of localization in the     presence of a network of sensors calibrated in space (θ,Δ). It being     known that, at the instant t_(k), the analytic relationship linking     the incidence (θ(t_(k),x,y,z), Δ(t_(k),x,y,z)) of the transmitter     with its position (x,y,z) is known, it is then possible, from the     incidence (θ(t_(k),x,y,z), Δ(t_(k),x,y,z)), to deduce the vector     a(t_(k),x_(m),y_(m),z_(m))=a(θ(t_(k),x,y,z), Δ(t_(k),x,y,z)) in     carrying out an interpolation of the calibration table (relative to     the calibrated antennas). It may be noted, however, that this method     is insensitive to a phase bias (owing to the vector correlation     criterion), -   These criteria also make it possible to take account of the phase     and amplitude of the components of a(θ,Δ). The method can therefore     be envisaged with networks of co-localized antennas with pattern     diversity. It must be noted that, in an airborne context, knowledge     of the altitude h of the aircraft reduces the computation of the     criterion in the search space (x,y) in assuming z=h. In the example     of the equations (6) and (10), the vector v_(k) (x,y,z) is written     as follows: $\begin{matrix}     {{v_{K}\left( {x,y,z} \right)} = \begin{bmatrix}     {\exp\left( {{j2\pi}\quad\frac{d}{\lambda}{\cos\left( {\theta\left( {t_{1},x,y,z} \right)} \right)}{\cos\left( {\Delta\left( {t_{1},x,y,z} \right)} \right)}} \right)} \\     {\exp\left( {{j2\pi}\quad\frac{d}{\lambda}{\cos\left( {\theta\left( {t_{K},x,y,z} \right)} \right)}{\cos\left( {\Delta\left( {t_{K},x,y,z} \right)} \right)}} \right)}     \end{bmatrix}} & (13)     \end{matrix}$

In this method, it is possible to envisage initializing the algorithm at K=K₀ and then recursively computing the criterion L_(k)(x,y,z). Under these conditions, L_(K)(x,y,z) is computed recursively as follows: $\begin{matrix} {{{L_{K + 1}\left( {x,y,z} \right)} = \frac{{{\alpha_{K + 1}\left( {x,y,z} \right)}}^{2}}{\beta_{K + 1}{\gamma_{K + 1}\left( {x,y,z} \right)}}}{where}{{\alpha_{K + 1}\left( {x,y,z} \right)} = {{\alpha_{K}\left( {x,y,z} \right)} + {b_{K + {1\quad m}}^{H}{b\left( {t_{K + 1},x,y,z} \right)}}}}{{\gamma_{K + 1}\left( {x,y,z} \right)} = {{\gamma_{K}\left( {x,y,z} \right)} + {{b\left( {t_{K + 1},x,y,z} \right)}^{H}{b\left( {t_{K + 1},x,y,z} \right)}}}}{\beta_{K + 1} = {\beta_{K} + {b_{K + {1\quad m}}^{H}b_{K + {1\quad m}}}}}} & (14) \end{matrix}$

The coefficients α_(K+1)(x,y,z)=α_(K)(x,y,z)γ_(K+1)(x,y,z)=γ_(K)(x,y,z), β_(K+1)=β_(K) are intermediate spectra enabling the computation of L_(k+1)(x,y,z).

When the vectors b(t_(K+1),x,y,z) and b_(km) have constant norms equal to ρ the relationship of recurrence of the equation (14) becomes: $\begin{matrix} {{{L_{K + 1}\left( {x,y,z} \right)} = \frac{{{\alpha_{K + 1}\left( {x,y,z} \right)}}^{2}}{\beta_{K + 1}{\gamma_{K + 1}\left( {x,y,z} \right)}}}{where}{{\alpha_{K + 1}\left( {x,y,z} \right)} = {{\alpha_{K}\left( {x,y,z} \right)} + {b_{K + {1\quad m}}^{H}{b\left( {t_{K + 1},x,y,z} \right)}}}}} & (15) \end{matrix}$

The method has been described hitherto in assuming that the vectors have fixed positions. It can easily be extended to the case of moving targets with a speed vector (v_(xm),v_(ym),v_(zm)) for which there is a model of evolution. In these conditions, the incidence of the m^(th) transmitter can be parametrized as follows: θ_(km)=θ(t _(k) , x _(m) −v _(xm) t _(k) , y _(m) −v _(ym) t _(k) , z _(m) −v _(zm) t _(k)) et Δ _(km)=Δ(t _(k) , x _(m) −v _(xm) t _(k) , y _(m) −v _(ym) t _(k) , z _(m) −v _(zm) t _(k))  (16) where (x_(m),y_(m),z_(m)) is the position of the transmitter at the instant to and (v_(xm),v_(ym),v_(zm)) are the components of the speed of the transmitter at the instant t₀. In these conditions, the vector b_(km) of the equation (9) is parametrized by (x_(m),y_(m),z_(m)) and (v_(xm),v_(ym),v_(zm)) as follows: b _(km) =b(t _(k) ,x _(m) ,y _(m) ,z _(m) ,v _(xm) ,v _(ym) ,v _(zm),)+w_(km)  (17)

Naturally, the criteria of localization L_(K) and L_(K)′ of the equations (11) and (12) are no longer parametrized solely by (x,y,z) but also by (v_(x),v_(y),v_(z)). The method then consists in maximizing the criterion L_(K)(x,y,z, v_(x),v_(y),v_(z)) as a function of the 6 parameters (x,y,z, v_(x),v_(y),v_(z)).

The method can be applied to a very large number of measurements. In this case, the method comprises a step of reduction of the numerical complexity of computation (which depends on the number of measurements) by reducing K. The method provides for the following processing operations to be performed on the elementary measurements:

-   decimation of the instants t_(k), in eliminating the neighboring     instants for which the development of the curve β_(m)(t_(k)) is not     significant, -   filtering (smoothing of the measurements which are the direction     vectors) and sub-sampling, -   the measurements are then merged on a defined duration (extraction     by association of direction vector to produce a synthesis     measurement).     Recapitulation of the Steps of the Method

The process of localization of several transmitters using K instants t_(k) can be summarized by the following steps:

-   Step No. 1: Identification of the vectors a_(km) for (1≦m≦M) at the     K instants t_(k) in applying for example a technique of source     separation, and source identification as described in the references     [2] [3]. -   Step No. 2: Association of the vectors a_(1m) up to a_(km) obtained     at the respective instants t₁ . . . t_(K) associated with the m^(th)     transmitter for 1≦m≦M in applying the steps ASE-1 to ASE-5 described     here above. -   Step No. 3: Initialization of the process at m=1. -   Step N° . 4: Conversion of the K vectors a_(km) into vectors b_(km)     as suggested by the equation (9). -   Step No. 5: Computation and maximization of the criterion     L_(K)(x,y,z) of the equation (11) to obtain the position     (x_(m),y_(m),z_(m)) of the m^(th) transmitter. -   Step No. 6: Incrementing m←m+1 and if m<M return to the step No. 3.     In order to refine the estimation of the position     (x_(m),y_(m),y_(m)) of the transmitters, the steps of the method may     be performed iteratively as follows: -   Step No. 7: Identification of the vectors b_(i) for (1≦i≦M) at the     instants t_(K+1) in applying for example a technique of source     separation and source identification as described in the references     [2] [3]. -   Step No. 8: For 1≦m≦M determining the indices i(m) in applying the     equation (4) and in using the vector a_(km) and the vectors b_(i)     for (1≦i≦M). -   Step No. 9: For 1≦m≦M the operation a_(k+1 m)=b_(i(m)) is performed. -   Step No. 10: For 1≦m≦M computation of the criterion L_(K+1)(x,y,z)     iteratively in using the equations (14) and (15) and minimization of     L_(K+1)(x,y,z) to obtain the position (x_(m),y_(m),z_(m)) of the     m^(th) transmitter. -   Step No. 11: If it is decided to go on, in order to achieve greater     precision and reduced ambiguity, then the method returns to the step     No. 7.     Exemplary Implementation of the Method

The simulations were made with a network of N=2 sensors aligned in the axis of the carrier with d/λ=3. Since d/λ=3 a method performing goniometry at the instants t_(k), it would be completely ambiguous and would not subsequently enable operations of triangulation for localizing the transmitter. In FIGS. 5, 6 and 7 corresponding to the criteria of localization for K=3, 7 and 16, the pseudo-spectrum L_(K)(x,y) to be maximized, enabling the position of the transmitter in the space (x,y) to be determined, is plotted. It being known that if the transmitter is situated in (x₀,y₀), then L_(K)(x₀,y₀)=1, it is deduced therefrom that the iso-level curves L_(K)(x,y)=0.99 characterize the width of the major lobe. Observing that the precision of localization depends on the width of this lobe, it is deduced therefrom, according to FIGS. 5, 6 and 7 that the greater the value of K, the greater will be the localizing precision.

REFERENCES

-   [1] R O. SCHMIDT. A signal subspace approach to multiple emitter     location and spectral estimation, November 1981 -   [2] J. F. CARDOSO, A. SOULOUMIAC, Blind beamforming for non-gaussian     signals, IEEE Proceedings-F, Vol. 140, N°6, pp. 362-370, December     1993. -   [3] P. COMON, Independent Component Analysis, a new concept ?,     Signal Processing, Elsevier, April 1994, Vol. 36, n°3, pp 287-314. 

1-9. (canceled)
 10. A method of localization of one or more sources, each source being in motion relative to a network of sensors, the method comprising the steps of separating the sources in order to identify the direction vectors associated with the response of the sensors to a source having a given incidence, associating direction vectors a_(1m) . . . a_(Km) obtained for the m^(th) transmitter and respectively at the instants t. . . t_(k), localizing the m^(th) transmitter from the associated vectors a_(1m) . . . a_(Km).
 11. The method according to claim 10, wherein associating step comprises: Step ASE-1: Initialization of the process at k=2. Step ASE-2: For 1≦m≦M determining the indices i(m) in using the relationship ${{d\left( {a_{k\quad m},b_{i{(m)}}} \right)} = {\min\limits_{1 \leq i \leq M}\left\lbrack {d\left( {a_{k\quad m},b_{i}} \right)} \right\rbrack}},$  the vector a_(k,m) and the vectors b_(i) identified at the instant t_(k+1) for (1≦i≦M), setting up a function β_(m)(t_(k))=d(a_(km), a_(om)), Step ASE-3: For 1≦m≦M performing the operation a_(k+1 m)=b_(i(m)), Step ASE-4: Incrementing k←k+1 and if k<K returning to the step ASE-1, Step ASE-5: Starting from the family of instants Φ={t₁<. . . <t_(k)} thus obtained, extracting the instants t_(i) which do not belong to a zone defined by the curve β_(m)(t_(k)) and a zone of tolerance.
 12. The method according to claim 10, wherein the localizaing step comprises: a normalized vector correlation L_(k)(x,y,z) maximizing in the space (x,y,z) of the position of a transmitter with ${L_{K}\left( {x,y,z} \right)} = \frac{{{b_{K}^{H}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}{v_{K}\left( {x,y,z} \right)}} \right)}$ with ${b_{K} = {\begin{bmatrix} b_{1m} \\ \vdots \\ b_{Km} \end{bmatrix} = {{v_{K}\left( {x_{m},y_{m},z_{m}} \right)} + w_{K}}}},{{v_{K}\left( {x,y,z} \right)} = {{\begin{bmatrix} {b\left( {t_{1},x,y,z} \right)} \\ \vdots \\ {b\left( {t_{K},x,y,z} \right)} \end{bmatrix}\quad{and}\quad w_{K}} = \begin{bmatrix} w_{1m} \\ \vdots \\ w_{Km} \end{bmatrix}}}$  where w_(K) is the noise vector for all the positions (x, y, z) of a transmitter.
 13. The method according to claim 12, wherein the vector b_(k) comprises a vector representing the noise, the components of which are functions of the components of the vectors a_(1m) . . . a_(Km).
 14. The method according to claim 12,wherein comprising: a step in which the matrix of covariance R=E[w_(K)w_(K) ^(H)] of the noise vector is determined and in that the following criterion is maximized: ${L_{K}^{\prime}\left( {x,y,z} \right)} = \frac{{{b_{K}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}R^{- 1}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}} \right)}$
 15. Method according to claim 14, wherein the evaluation of the criterion L_(K)(x,y,z) and/or of the criterion L_(K)′(x,y,z) is recursive.
 16. The method according to claim 14, wherein it comprises a step of comparison of the maximum values with a threshold value.
 17. The method according to claim 11, wherein the value of K is initially fixed at K₀.
 18. The method according to claim 10, wherein the transmitters to be localized are mobile and in that the vector considered is parameterized by the position of the transmitter to be localized and the speed vector. 